Trainers are NPCs that have their own Mortys that you can battle against. Take the portal at the Citadel of Ricks and you will find them scattered around the dimensions. Every dimension has one Rick and multiple non-Rick trainers. You can also battle Council Members in the Council Chamber in the Citadel of Ricks (when you have enough badges).

Updated to Game Version 2.3.5, Asset Version 37 (February 19th, 2018).